When calculating with matrices, we make use of the fact that if \(U(k)\subset U\)  is dense (e.g. if \(k\)  is perfect [1]}), then a map factors through the trivial group if and only if it maps to the identity on \(U(k)\) .
Since \(R={R}_u(G_{\bar{k}})\)  descends to \(R^{\prime }:={R}_{u,k^{\prime }}(G_{k^{\prime }})\)  whose \(k^{\prime }\) -points are dense (by the fact that \(k^{\prime }\)  is separably closed) we have \(e(R)\)  is the smallest \(s\)  such that \(p^s\)  kills \(R^{\prime }(k^{\prime })\) .
